Research Talk

Applied Probability

Tenure Track Interview

Joost Jorritsma

slides.com/joostjor/leiden-interview

Internet: a growing network of routers and servers

~1969: 2 connected sites

Time

~1989: 0.5 million users

~2023: billions of devices

[Faloutsos, Faloutsos & Faloutsos, '99]:
- Short average distance:
  Quick spread of information

~1999: 248 million users

Distance evolution in a growing network

1999

$\mathrm{dist}_{\color{red}{'99}}(u_{'99}, v_{'99}) = 4$

2005

$\mathrm{dist}_{{\color{red}'05}}(u_{'99}, v_{'99}) = 3$

2024

$\mathrm{dist}_{{\color{red}'24}}(u_{'99}, v_{'99}) = 2$

21 possible networks

Attachment rule:

Prefer connecting to high-degree vertices, $\tau$ : tail of power-law degree distribution

2005

$\phantom{\mathrm{dist}_{{\color{red}'05}}(u_{'99}, v_{'99}) = 3}$

Distance evolution: hydrodynamic limit

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$ , then

$\sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\overset{\mathbb{P}}\longrightarrow 0.$

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $\phantom{t'=T_t(a):=t\exp\big(\log^a(t)\big)}$ for $\phantom{a\in[0,1]}$ , then

$\phantom{\sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\overset{\mathbb{P}}{\longrightarrow} 0.}$

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$ , then

$\phantom{\sup_{a\in[0,1]}} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} \phantom{- (1-a)\frac{4}{|\log(\tau-2)|}}\right|\phantom{\overset{\mathbb{P}}\longrightarrow 0.}$

```
Dynamics in PAMs.
```

Generalization with edge weights: 
random transmission times

Novelties

```
Fast spreading among influentials;
```

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$ , then

$\sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\phantom{\overset{\mathbb{P}}\longrightarrow 0.}$

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume $\tau<3$ .
Let $t'=T_t(a):=t\exp\big(\log^a(t)\big)$ for $a\in[0,1]$ , then

$\phantom{\sup_{a\in[0,1]}} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\phantom{\overset{\mathbb{P}}\longrightarrow 0.}$

Information diffusion in random graphs

Distances

Component sizes

Intervention strategies

FINAL SIZE

Hyperbolic random graph

Scale-free percolation

Long-range percolation

Bond percolation on $\mathbb{Z}^d$

\exp\big(-\Theta(\sqrt{n})\big).

\exp\big(-\Theta(\sqrt{n})\big).

Theorem ( $\mathbb{Z}^d$ -like graphs)

[Lebowitz & Schonmann '88; Gandolfi '89; Grimmett, Marstrand '90;

Kesten, Zhang '90; Alexander, Chayes, Chayes, Newman '90; Pisztora '96;
Cerf '97; Contreras, Martineau, Tassion '2024; ...]

Lower tail:
Surface tension drives too small cluster

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)= \phantom{\Theta\big(n^{-I(\varepsilon)}\big)}

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)= \phantom{\Theta\big(n^{-I(\varepsilon)}\big)}

Upper tail:
Large clusters are very unlikely

\exp(-\Theta(n)\big).

\exp(-\Theta(n)\big).

Figure by Tobias Muller

Hyperbolic random graph

Scale-free percolation

Long-range percolation

Bond percolation on $\mathbb{Z}^d$

\exp\big(-\Theta(\sqrt{n})\big).

\exp\big(-\Theta(\sqrt{n})\big).

Theorem ( $\mathbb{Z}^d$ -like graphs)

[Lebowitz & Schonmann '88; Gandolfi '89; Grimmett, Marstrand '90;

Kesten, Zhang '90; Alexander, Chayes, Chayes, Newman '90; Pisztora '96;
Cerf '97; Contreras, Martineau, Tassion '2024; ...]

Lower tail:
Surface tension drives too small cluster

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)= \phantom{\Theta\big(n^{-I(\varepsilon)}\big)}

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)= \phantom{\Theta\big(n^{-I(\varepsilon)}\big)}

Upper tail:
Large clusters are very unlikely

\exp(-\Theta(n)\big).

\exp(-\Theta(n)\big).

Question:

Long edges and high-degree vertices,

do they matter?

\exp\big(-\Theta(n^\zeta)\big).

\exp\big(-\Theta(n^\zeta)\big).

Theorem [J., Komjáthy, Mitsche, '24+]
We find explicit $\zeta\in[1/2,1)$ , $\theta\in(0,1)$ s.t.

Lower tail: small final size
If enough long edges or no hubs

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=

Novelties

Reversed discrepancy: large outbreak likelier than small.

Long edges can beat surface tension: any         ;

\zeta\!\in\![1/2, 1)

\zeta\!\in\![1/2, 1)

  governs second-largest cluster, and cluster of 0

\zeta

\zeta

Techniques: probability, combinatorics, optimization.

What is the influence of long edges and high-degree vertices?

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)= \phantom{\Theta\big(n^{-I(\varepsilon)}\big)}

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)= \phantom{\Theta\big(n^{-I(\varepsilon)}\big)}

Upper tail: large final size
If hubs are present, we find rate funtion $I(\varepsilon)$ :

What is the influence of long edges and high-degree vertices?

\Theta\big(n^{-I(\varepsilon)}\big).

\Theta\big(n^{-I(\varepsilon)}\big).

1 . 1

Research Talk Applied Probability Tenure Track Interview Joost Jorritsma slides.com/joostjor/leiden-interview

Research Talk

Joost Jorritsma

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